Holographic Entanglement Entropy and Renormalization Group Flow

Abstract

Using holography, we study the entanglement entropy of strongly coupled field theories perturbed by operators that trigger an RG flow from a conformal field theory in the ultraviolet (UV) to a new theory in the infrared (IR). The holographic duals of such flows involve a geometry that has the UV and IR regions separated by a transitional structure in the form of a domain wall. We address the question of how the geometric approach to computing the entanglement entropy organizes the field theory data, exposing key features as the change in degrees of freedom across the flow, how the domain wall acts as a UV region for the IR theory, and a new area law controlled by the domain wall. Using a simple but robust model we uncover this organization, and expect much of it to persist in a wide range of holographic RG flow examples. We test our formulae in two known examples of RG flow in 3+1 and 2+1 dimensions that connect non–trivial fixed points.

A useful probe of the properties of various field theories that has
received increased interest in recent times is the entanglement
entropy, with applications being pursued in diverse areas such as
condensed matter physics, quantum information, and quantum
gravity. One of the main motivators, in the context of strongly
coupled field theories (perhaps modeling novel
new phases of matter), is that the entanglement entropy may well act
as a diagnostic of important phenomena such as phase transitions, in
cases where traditional order parameters may not be available.

Within a system of interest, consider a region or subsystem and call it
A, with the remaining part of the system denoted by
B. A definition of the entanglement entropy of
A with B is given by:

SA=−TrA(ρAlnρA),

(1)

where ρA is the reduced density matrix of A given by tracing over the degrees of freedom of B,
ρA=TrB(ρ),
where ρ is the density matrix of the system. When the system is in a pure state, i.e.,ρ=|Ψ⟩⟨Ψ|,
the entanglement entropy is a measure of the entanglement between the
degrees of freedom in A with those in B.

It is of interest to find ways of computing the entanglement entropy
in various strongly coupled systems, in diverse dimensions, and under
a variety of perturbations, such as the switching on of external
fields, or deformations by relevant operators. A powerful tool for
studying such strongly coupled situations is gauge/gravity duality,
which emerged from studies in string theory and M-theory. The best
understood examples are the conjectured AdS/CFT correspondence and its
numerous deformations
[1, 2, 3, 4] (See e.g., ref.[5] for
an early, but still very useful, review.) There has
been a great deal of activity for over a decade now, applying these
tools to strongly coupled situations of potential interest in
condensed matter and nuclear physics, for example. Fortunately, there
has been an elegant proposal[6, 7] for how to compute the entanglement
entropy in systems with an Einstein gravity dual (or, more generally, a string or
M–theory dual in the large N limit and large t’ Hooft limit), which provides a new way to
calculate the entanglement entropy using geometrical techniques (for a
review see ref.[8]). In an asymptotically Anti–de
Sitter (AdS) geometry, consider a slice at constant AdS radial
coordinate z=a. Recall that this defines the dual field theory
(with one dimension fewer) as essentially residing on that slice in
the presence of a UV cutoff set by the position of the slice. Sending
the slice to the AdS boundary at infinity removes the cutoff (see
ref. [5] for a review). On our z=a slice, consider
a region A. Now find the minimal–area surface
γA bounded by the perimeter of A and
that extends into the bulk of the geometry. (Figure 1 shows examples of the arrangement we will consider in this paper.)
Then the entanglement
entropy of region A with B is given by:

SA=Area(γA)4GN,

(2)

where GN is Newton’s constant in the dual gravity
theory.

(a)The strip.

(b)The disc.

Figure 1: Diagrams of the two shapes we will consider for region A. This is the case of AdS4, and here, z denotes the radial direction in AdS4. In one dimension higher we will generalize these shapes to a box and a round ball, and in one dimension fewer, we will consider an interval.

This prescription for the entropy coincides nicely with
various low dimensional computations of the entanglement entropy, and
has a natural generalization to higher dimensional theories. Note
that there is no formal derivation of the prescription. Steps have
been made, such as in refs. [9, 10], but
they are not complete. However, there is a lot of evidence for the
proposal. See e.g.,
refs.[11, 12, 13, 14, 15, 16]. A
review of several of the issues can be found in
ref.[8]. Further progress has been made recently
in ref. [17].

In this paper we shall assume that this holographic prescription does
give the correct result for the entanglement entropy in systems with
gravity duals, and proceed to examine the interesting question of how
the entanglement entropy behaves when a system is perturbed by an
operator that triggers a Renormalization Group (RG) flow. For
simplicity, we will work with flows that connects two conformal field
theories, and we will consider (for concreteness) a four dimensional
example and one in three dimensions. Such examples are extremely natural to study using holographic duality
since (at large N) it is possible to find geometries that represent the full flow
from the maximally supersymmetric theory to theories with fewer
super symmetries. (This was first proposed in
refs.[18, 19], and several examples
have since been found.) Flow between field theory fixed points correspond to flows between
fixed points of the supergravity scalar potential. The examples we will study begin with the four
dimensional case of the
flow[20, 21, 22] to the
Leigh–Strassler point[23, 24], which
results from giving a mass to one of the N=1 chiral
multiplets that make up the N=4 Yang Mills gauge multiplet.
We then continue with the three dimensional generalization of it
discussed in ref. [25]. The gravity dual of the four
dimensional flow connects AdS5×S5 at the r=+∞ extreme
of a radial coordinate r to AdS5×M5 at r=−∞,
where the space M5 results from squashing the S5 along
the flow. There are two of the 42 supergravity scalars switched on at
the latter endpoint, and correspondingly the characteristic radius of
the AdS5 in the IR is larger than that of the UV theory: The gravity dual for the three dimensional
flow has related structures, this time connecting an AdS4×S7
UV geometry to an AdS4×M7 in the IR, where M7 results from squashing the S7 along the flow.

Before studying the specific examples, however, we step back and try
to anticipate some of the key physics that we should expect from the
entanglement entropy in this type of situation, more generally. Generically, holographic
RG flow involves a flow from one dual geometry in the UV to another in
the IR, separated by an interpolating region that can be thought of as
a domain wall separating the two regions. The key to understanding the
behaviour of the content of the holographic entanglement entropy
formula is to then understand how the computation incorporates the
structure of the domain wall, and how the field theory quantities it
extracts are encoded. To anticipate how to mine this information, we
do an analytic computation of the proposed entanglement
entropy (2) in an idealized geometry
given by a sharp domain wall separating two AdS regions with different
values for the cosmological constant. Working in various dimensions
(AdS5, AdS4, and AdS3, pertaining to flows in four, three,
and two dimensional field theories), we find a fascinating and
satisfying structure, seeing how the entanglement entropy tracks the
change in degrees of freedom under the flow, and several other features. We
expect that these features will be present in a wide range of
examples, and we confirm our results in the examples mentioned above.

The outline of this paper is as follows. In section 2 we carry out the
study of the entanglement entropy in the presence of the idealized (i.e., sharp domain wall)
holographic RG flow model, and discover how the physics is organized
in the results. Then, ready to study examples, we review the four
dimensional Leigh–Strassler RG flow of interest, and its dual AdS5
flow geometry in section 3. We explicitly solve (numerically) the
non–linear equations that define the geometry and scalars in the
interpolating dual supergravity flow. We then compute the
entanglement entropy and extract the physics, comparing to our
predictions from section 2. Section 4 presents the analogous studies
for the three dimensional field theory, with the AdS4 dual flow
geometry. We end with a
discussion in section 5.

As mentioned in the introduction, the generic holographic RG flow
involves a flow from one dual geometry in the UV to another in the IR,
separated by an interpotating domain wall.
In all examples, understanding the behaviour of holographic
entanglement entropy, as proposed in
equation (2), requires us to understand
how the area formula incorporates the structure of the domain wall in terms of field theory quantities. So we start by
doing an analytic computation in an idealized geometry given by a
sharp domain wall in AdS. In general, the location of the wall, and
its thickness, are determined by field
theory parameters corresponding to the details of the relevant
operator - for example, in the case of the Leigh–Strassler flow and
its generalization we later study, the detail in question is the bare
value of the mass given to the chiral multiplet. A sharp
domain wall is of course not a supergravity solution, and falls
somewhat outside the usual supergravity duality to any (large N)
theory, but nevertheless is a clean place to start to capture how the
physics is organized. We expect it to capture a great deal of the
key physics of holographic RG flow, as regards how the entanglement
entropy formula works.

We use the following background metric:

ds2=e2A(r)(−dt2+d→x2)+dr2,

(3)

with

A(r)={r/RUV,r>rDWr/RIR,r<rDW.

(4)

Here −∞<r<+∞, and →x is either four, three, or two
coordinates (the spatial coordinates of the dual field theory),
depending upon whether we are in AdS5, AdS4, or AdS3, the
cases we will consider. Also, RIR>RUV. The length scale of AdS
on either aide of the wall is set by RUV in the UV at r>rDW
and RIR in the IR at r<RDW.

2.1 The Ball and AdS5.

We begin by studying a region A in the three spatial
dimensions which is a round ball of radius ℓ. Using a radial
coordinate ρ in the spatial dimensions, the area of the surface,
γ, that extends into the bulk is given by:

Area=4π∫ℓ0dρρ2e3A(r)(1+e−2A(r)r′(ρ)2)1/2,

(5)

where the function r(ρ) defines the enbedding. We can calculate the equations of motion that result from minimizing this “action,” and we find that the solution is given by:

where rDW is the position of the domain wall in the AdS radial
direction, and ρDW is the spatial radial position where
r(ρDW)=rDW and is given by:

ρ2DW=ℓ2+ϵ2−R2UVe−2rDWRUV.

(7)

Note that, rather than integrating out to r=+∞, we integrate
out to large positive radius rUV, defining our UV cutoff, with
small ϵ defined by:

rUV=−RUVln(ϵRUV).

(8)

Note that with the solution given for ρ(ρ), we are assuming
that ℓ is larger than a critical radius ℓcr such that our
surface extends past the doman wall into the second AdS region. The
critical radius ℓcr is given by setting ρDW=0 in the
above:

ℓ2cr=R2UVe−2rDWRUV−ϵ2.

(9)

Substituting the solution back into equation (5), we can
analytically calculate the area of our minimal surface, γA, and hence the entanglement entropy via
equation (2). This gives a long
expression that we will not display here. For our purposes it is
enough to first expand the area for small ϵ:

There are a number of notable features of this expression. First, we
see the results from pure AdS5 in the first line. There, we see
the usual UV divergent terms and the ℓ–independent constant
that results from the fact that the ball preserves some of the
conformal invariance of AdS5. Second, the terms that have RIR as
coefficients (the last two lines) always have ℓ2 appearing in
the combination:

~ℓ2=ℓ2−R2UVe−2rDWRUV=ℓ2−ℓ2cr+O(ϵ2).

(13)

We are tempted to interpret this ~ℓ as the effective ball
radius as seen in the IR, as opposed to the simple ℓ seen in the
UV. Furthermore, the combination:

~ϵ=RIRe−rDWRIR

(14)

appears in a manner analogous to how the UV cut-off ϵ
appears. (This might not be clear in our ϵ expansion of the
above equation. One way to see that it does appear as ϵ
does is to look at equation (6)). Now ~ϵ
is not necessarily small, but we will see that it is useful to think
of it as the cut–off in the IR theory. With these observations in
mind, we rewrite the last three lines of equation (11) as
follows:

Let us focus on the terms proportional to R3IR. If we expand
these terms assuming that ~ϵ/~ℓ<<1,
i.e., the effective length in the putative IR theory is larger than the IR
cutoff, which also means that the length is such that the surface
extends very far past the wall into the IR AdS space, we get:

R3IR2[~ℓ2~ϵ2+ln(~ϵ~ℓ)]+R3IR4[1−2ln(2)]+O(~ϵ/~ℓ).

(16)

Pleasingly, this is exactly the result we would have obtained if we
were purely in the IR theory!

So far therefore, we have seen how the entanglement entropy formula
encodes key behaviours of both the UV and the IR theories, in terms of
the appropriate scales, ϵ/ℓ and ~ϵ/~ℓ. The boundary of AdS5 at r=+∞ is
the UV region and the quantities of the UV theory appear
accordingly. From the point of view of the IR theory, the domain wall
acts (for ~ϵ/~ℓ small) as the effective UV
region, with ~ϵ/~ℓ acting as the effective
regulator.

We are left with understanding the first two terms in
equation (15).
These two terms mix the properties of the UV and the IR regions, and
are more subtle. We associate them with the region around the domain
wall, which connects the UV and IR regions (through and abrupt change
in our idealized example). It is prudent to try to understand the role
of these terms toward the end of the flow, and so we do a large ℓ
expansion of them, giving:

So we see that these terms give contributions very analogous to our UV
and IR results, where here the reference scale is played by
~ℓcr. (Note that the constant term is actually
different than the UV and IR constant terms’ form.) At fixed
ϵ or ~ϵ, we may think of this as a new set of
divergences.

Now that we have an understanding of the contributions of the various
pieces to the area, we combine everything together again and consider
the large ℓ (and small ϵ) expansion:

The first key result here is that we no longer have a ln(ℓ)
scaling associated with the UV theory. The remaining ln(ℓ)
dependence has a coefficient that is only associated with the IR
theory and that is independent of the domain wall. In a non–RG flow scenario, the coefficient of such a term is determined by the central charge of
the theory (see e.g., refs.[7, 26]), but here we see that the coefficient has shifted from its UV
value (associated with the UV central charge) to its IR value (associated with the IR central charge).
The second thing to note is that the area law associated with the UV
cut–off (the first ℓ2 term) is joined by a second area law. Its
coefficient is sourced by the details of the domain wall. For
clarity, we display this term here:

ℓ22⎛⎝R3IR~ϵ2−R3UV~ℓ2cr⎞⎠=ℓ22(RIRe2rDWRIR−RUVe2rDWRUV).

(19)

We expect this new area law to be a robust feature of RG flow
geometries, but anticipate that the coefficient’s precise form will be
different as we move away from the thin wall limit we are in here.
The above result predicts that the coefficient grows more positive as
rDW is pushed to the UV. In realistic RG flows, while the domain
wall position and sharpness cannot be varied arbitrarily, it is
expected to get thinner toward the UV and so at least in that regime
we should recover positivity.
Finally, the constant terms in the last line of
equation (18) are a mixture of both the UV, IR, and
domain wall physics.

2.2 The Disc and AdS4.

We can repeat the same procedure for AdS4, pertaining to RG flows
in 2+1 dimensional theories. As our system A we consider a
circular disc of radius ℓ. The solution for the surface embedding
are exactly as in equation (6). We can calculate the minimal
area and expand for small ϵ to get:

We see the reappearance of many of the key players that we saw in the
AdS5 case, such as ~ℓ, ~ℓcr and
~ϵ, appearing in similar types of term. For ℓ=~ℓcr, we recover the pure UV result (proportional to
ℓ/ϵ) and also the constant −R2UV, the
constant ensured by the fact that the disc preserves some conformal
invariance, as expected. For large ℓ, we have:

Area2π=R2UVℓϵ−R2UVℓ~ℓcr+R2IRℓ~ϵ−R2IR+O(ϵ,1/ℓ).

(21)

So in the AdS4 case, the constant term shifts from its UV result to
its IR result −R2IR. Again, in addition to the usual UV area law (proportional to
ℓ/ϵ), we have a new area law controlled by the domain wall:

ℓ(R2IR~ϵ−R2UV~ℓcr),

(22)

which should be compared to the example from AdS5 in
equation (19). The same comments we made for the
new area law there apply here: It is not necessarily positive, but we
expect it to get more positive as the domain wall is sent to the UV,
where generically it gets thinner.

2.3 The Case of AdS3.

Next we consider the case of AdS3, pertaining to flows in 1+1
dimensions. We use a spatial interval of length 2ℓ for our region
A. The area is given by:

where ~ℓcr and ~ϵ are defined in equations (12) and (14) respectively. So again we see that the universal coefficient (in front of the
natural logarithm) becomes the IR factor in the large ℓ limit.
The IR cutoff replaces the UV cutoff just as observed before.

2.4 The Strip and AdS4.

We next consider an area A that is a strip in AdS4, to
compare our results for the disc. We take the strip to be of finite
width ℓ in the x direction, and of length L in the remaining
direction, which will be taken to be large, making an infinite
strip. The area is given by:

Area=2L∫ℓ/20dxe2A(r)√1+e−2A(r)r′(x)2.

(25)

Since there is no explicit dependence on x, there is a constant of
motion in the dynamical problem associated to minimizing the area.
However, we must be careful since the constant of motion on either
side of the domain wall is not the same:

e2A(r)√1+e−2A(r)r′(x)2=⎧⎪⎨⎪⎩e2r∗RUV,r>rDWe2r∗RIR,r<rDW.

(26)

On the IR side, the constant is simply given by r′(x)=0, which
occurs at a radial position we will denote as r∗. The constant
on the UV side is determined by asking that r′(x)=0 when r∗=rDW, which is the critical situation before our embedding enters
the IR AdS. We can in turn calculate the area and length in terms
of r∗ and expand for small ϵ:

So we see that that the constant term here is exactly the new area
law’s coefficient that we saw in the disc case, in equation (22). Again, far enough in the UV, for large enough mass, our analysis suggests that this coefficient is positive.

2.5 The Box and AdS5.

Returning to AdS5, we consider for region A a box in
AdS5, in order to compare to the round ball we studied before.
Here the finite width is again ℓ and the two other sides are of
length L, which we again take to be large. The computation proceeds
in a similar way. The area gives:

3.1 The Holographic Dual Gravity Background

In field theory terms, the RG flow is defined by an N=1
supersymmetric deformation of the N=4 supersymmetric Yang
Mills theory given by introducing a mass term for one of the chiral
multiplets.This relevant deformation causes the N=4 theory to flow
to an N=1 fixed point in the IR called the
Leigh–Strassler fixed
point [22, 23, 24]. For the SU(N)
theory at large N, there is an holographic dual of this
physics [20], represented by a flow between two five
dimensional anti–de Sitter (AdS5) fixed points of N=8
gauged supergravity in five dimensions. One point has the maximal
SO(6) symmetry, and the other has SU(2)×U(1), global
symmetries of the dual field theories.
The relevant five dimensional gauged supergravity action is
[20, 21, 27]:

S=116πG5∫d5x√−g(R−2(∂χ)2−12(∂α)2−4P),

(33)

with

P=12R2(16(∂W∂α)2+(∂W∂χ)2)−43R2W2,

(34)

where the superpotential W is given by:

W=14ρ2(cosh(2χ)(ρ6−2)−(3ρ6+2)),

(35)

with ρ=exp(α). The scalar field χ is dual to an
operator of dimension three in the field theory while the scalar field
α is dual to a dimension two operator:

α:4∑i=1Tr(ϕiϕi)−26∑i=5Tr(ϕiϕi),χ:Tr(λ3λ3+φ1[φ2,φ3])+h.c.,

(36)

where φk=ϕ2k−1+iϕ2k, k=1,…,3. Here ϕi (i=1,…,6) are the six scalars in the N=4 multiplet, and the λk (three of that adjoint
multiplet’s four fermions) are N=1 partners of the
φk, forming the three chiral multiplets. This combination of
operators is exactly what is needed to reproduce the deformation.
The geometry in five dimensions, of domain wall form, can be
parametrised in the following manner:

ds21,4=e2A(r)(−dt2+dx21+dx22+dx23)+dr2.

(37)

The supergravity equations of motion yield the following flow
equations:

dαdr

=

eα6R∂W∂α=16R(e6α(cosh(2χ)−3)+cosh(2χ)+1e2α),

dχdr

=

1R∂W∂χ=12R((e6α−2)sinh(2χ)e2α),

dAdr

=

−23RW=−16Rcosh(2χ)(e6α−2)−(3e6α+2)e2α.

(38)

In these coordinates, the UV is at r→+∞ and the IR is at
r→\-∞, as in earlier sections. In either limit, the right
hand side of the first two equations vanish, and the scalars run to
specific values (α=0,χ=0 at one end, α=16ln2,χ=12ln3 at the other), while A(r) becomes rR in
the UV and 25/33rR in the IR, defining an
AdS5 in each case, and hence a conformally invariant dual field
theory at each end. This is a fat, smooth version of our simple thin
domain wall model of the previous sections. Here RUV=R and
RIR=3R/25/3.

To study the UV behavior of the fields, we find it convenient to
define a coordinate ~z given by:

~z=e−r/R,

(39)

and we find the asymptotic behavior of the fields near ~z=0 (UV AdS boundary):

To study the IR behavior of the fields, we define a coordinate ~u given by:

~u=eλr/R,

(42)

where λ=25/3(√7−1)3. The
asymptotic (near ~u=0) behavior in the IR is given by:

χ(~u)

=

12ln(3)+~ub0+O(~u2),

α(~u)

=

16ln(2)+~u(√7−16)b0+O(~u2),

A(~u)

=

1√7−1ln(~u)+B0+O(~u2).

(43)

3.2 Numerically Solving for the Flow

To solve the flow numerically (as we will need to do in order to
compute the entanglement entropy), it is convenient to work with a
coordinate:

x=~z2,

(44)

and employ a shooting method to solve the equations. To shoot from the
IR we take xmax=106, and towards the UV we take x=ϵ, where we use ϵ=10−12. To get good numerical
stability, we define new fields (β,η),

α(x)=xβ(x),χ(x)=x1/2η(x),A(x)=−12ln(x)+a(x),

(45)

such that near the AdS boundary, the leading behavior of these fields is given by:

η(x)=a0+O(x),β(x)=a1−13a20ln(x)+O(x),a(x)=A0−16a20x+O(x2).

(46)

In the IR, we use the results in equation (3.1) (up to
O(~u4)) as our shooting conditions (we have the freedom of
choosing the parameter b1, which is always less than zero). We can
then extract the values of a0 and a1 at the AdS boundary for our
solution. Note that the choice of B0 will determine the choice of
A0.
We can choose to eliminate the constant A0 by appropriately
rescaling our coordinate ~z. In particular, if we solve our
equations with B0=0, we can extract the constant A0, and then
simply perform the following transformation to eliminate it from our
metric:

~z→eA0~z.

(47)

Finally, as indicated in ref. [21], there are
constants of the motion regardless of the choice of b1 (and
subsequently, the choice of (a0,a1)), given by: